本题是SAT数学，请分析图像、给出解题步骤、并给出答案---**Question Number:** 22.
**Diagram Description:**
* Type: Geometric figure showing intersecting lines.
* Elements:
* Two horizontal lines labeled 'a' and 'b', stated as parallel lines. Line 'a' is above line 'b'.
* Three other lines labeled 'c', 'd', and 'e' intersect lines 'a' and 'b'.
* Line 'c' intersects 'a' and 'b' from top-left to bottom-right.
* Line 'd' intersects 'a' and 'b' from top-right to bottom-left.
* Line 'e' intersects 'a' and 'b' from top-left to bottom-right, positioned to the right of line 'c'.
* Angles are labeled:
* An angle above line 'a' and to the left of line 'c' is labeled w°.
* An angle above line 'a' and to the right of line 'd' is labeled v°.
* An angle below line 'a' and to the left of line 'd' is labeled x°.
* An angle above line 'b' and to the left of line 'd' is labeled y°.
* An angle below line 'b' and to the right of line 'e' is labeled z°.
**Question Stem:**
In the figure, parallel lines a and b are intersected by lines c, d, and e. If z = 71, y = 114, and v < z, which statement about x and w must be true?
**Options:**
A. x < w
B. x > w
C. x = w
D. x + w = 90
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Let's analyze this SAT geometry problem step by step. We have two parallel lines a and b intersected by three transversals c, d, and e. We're given that angle z equals 71 degrees, angle y equals 114 degrees, and angle v is less than angle z. We need to find the relationship between angles x and w.
First, let's find angle x using the properties of parallel lines. When a transversal intersects two parallel lines, consecutive interior angles are supplementary. Angles x and y are consecutive interior angles formed by transversal d. Therefore, x plus y equals 180 degrees. Since y equals 114 degrees, we can solve: x plus 114 equals 180, so x equals 66 degrees.
Now let's analyze the triangle formed by the intersection of lines c and d above line a. This triangle has vertices at point P where lines c and d meet, point A-c where line c meets line a, and point A-d where line d meets line a. The interior angles are: w degrees at A-c, 114 degrees at A-d, and alpha degrees at P. Since the sum of angles in any triangle equals 180 degrees, we have w plus 114 plus alpha equals 180. Therefore, alpha equals 66 minus w.
For the triangle to exist, all angles must be positive. Since alpha equals 66 minus w, we need alpha to be greater than zero. This gives us the constraint that 66 minus w must be greater than zero, which means w must be less than 66 degrees. Since we found that x equals 66 degrees, we can conclude that w is less than x, or equivalently, x is greater than w.
Let's summarize our solution. We found that x equals 66 degrees using consecutive interior angles. We determined that w must be less than 66 degrees from the triangle constraint. Therefore, x is greater than w. Looking at the answer choices: A says x is less than w, which is false. B says x is greater than w, which is correct. C says x equals w, which is false. D says x plus w equals 90, which is not necessarily true. The answer is B: x is greater than w.